Stir the shallot, garlic, white pepper, and vermouth together in a small bowl; set aside. Cut the rind off the top of the brie cheese, leaving a 1/4-inch lip around the edge. Place the brie cheese onto a baking sheet, and spoon the shallot mixture over top. Fold the phyllo sheets in half to make square pieces. Wrap the brie with the phyllo squares, tucking the edges underneath of the brie cheese. Trim the corners if needed to avoid tucking too much pastry underneath. You want complete enclosure, but without too much overlap. Brush the top with melted butter to make the pastry brown in the oven.
Bake in the preheated oven until the phyllo has turned golden brown and the brie cheese is very soft, about 20 minutes. Serve warm.